To predict the distribution of such defects within a single crystal, a method has been known which consists of applying an overall heat conduction simulation to a silicon single crystal 4 pulled up from a pulling-up machine 1 according to the CZ method as shown in FIG. 11. The method comprises the steps of predicting the distribution of temperatures within a silicon melt 2 by working on the heat conductivity of silicon melt 2 making allowance for the structural elements of a hot zone of the pulling-up machine 1 and the pulling-up speed of silicon single crystal 4, applying a mesh over the silicon single crystal 4 and determining, for each mesh, its coordinates and temperature from the internal temperature distribution, and solving a diffusion equation based on the diffusion coefficients of silicon atoms and air holes within an inter-lattice space of silicon single crystal 4 and boundary conditions using a computer, thereby determining the distribution of silicon atoms and air holes in the inter-lattice space. According to this simulation method, members constituting the hot zone are covered with a mesh for modeling. What is noteworthy is that each of the mesh applied to silicon melt 2 has a rather gross area or a square of about 10 mm, in order to reduce the time of computation.
A method for producing a silicon single crystal by the CZ method has also been known which consists of feeding, as input, the operation condition of the machine and temperature distribution within the crucible to a computer, thereby allowing the computer to predict, by simulation, the sizes and densities of grow-in defects due to air holes which might develop in the single crystal at the given growth speed of crystal based on a known relationship, and adjusting, based on the above simulation result, the machine such that the growth speed of crystal is kept at a specified level so as to make the sizes and densities of grow-in defects which might develop within the single crystal during its formation equal to or below predetermined thresholds (see, for example, Patent literature 1). This method uses an overall heat conduction analysis program to simulate the heat distribution within a crucible and determines the temperature gradient along a vertical axis passing through the center of a growing single crystal.
Patent literature 1: Japanese Unexamined Patent Application Publication No. 2002-145696